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Daunis i Estadella, Josep
MartÃn FernÃ¡ndez, Josep Antoni 

Universitat de Girona. Departament dâ€™InformÃ tica i MatemÃ tica Aplicada  
Egozcue, Juan JosÃ©
DÃaz Barrero, JosÃ© Luis Pawlowsky Glahn, Vera 

A joint distribution of two discrete random variables with finite support can be displayed as a two way table of probabilities adding to one. Assume that this table hasn rows and m columns and all probabilities are nonnull. This kind of table can beseen as an element in the simplex of n Â· m parts. In this context, the marginals areidentified as compositional amalgams, conditionals (rows or columns) as subcompositions. Also, simplicial perturbation appears as Bayes theorem. However, the Euclideanelements of the Aitchison geometry of the simplex can also be translated into the tableof probabilities: subspaces, orthogonal projections, distances.Two important questions are addressed: a) given a table of probabilities, which isthe nearest independent table to the initial one? b) which is the largest orthogonalprojection of a row onto a column? or, equivalently, which is the information in arow explained by a column, thus explaining the interaction? To answer these questionsthree orthogonal decompositions are presented: (1) by columns and a rowwise geometric marginal, (2) by rows and a columnwise geometric marginal, (3) by independenttwoway tables and fully dependent tables representing rowcolumn interaction. Animportant result is that the nearest independent table is the product of the two (rowand column)wise geometric marginal tables. A corollary is that, in an independenttable, the geometric marginals conform with the traditional (arithmetic) marginals.These decompositions can be compared with standard loglinear models.Key words: balance, compositional data, simplex, Aitchison geometry, composition,orthonormal basis, arithmetic and geometric marginals, amalgam, dependence measure,contingency table Geologische Vereinigung; Institut dâ€™EstadÃstica de Catalunya; International Association for Mathematical Geology; CÃ tedra LluÃs SantalÃ³ dâ€™Aplicacions de la MatemÃ tica; Generalitat de Catalunya, Departament dâ€™InnovaciÃ³, Universitats i Recerca; Ministerio de EducaciÃ³n y Ciencia; Ingenio 2010. 

http://hdl.handle.net/2072/14734  
eng  
Universitat de Girona. Departament dâ€™InformÃ tica i MatemÃ tica Aplicada  
Tots els drets reservats  
Geometria dâ€™Aitchison
Probabilitats 

Compositional analysis of bivariate discrete probabilities  
info:eurepo/semantics/conferenceObject  
Recercat 